We study the two-dimensional (2D) t−J model (and its extensions) in order to understand the hole dynamics in the 2D CuO₂ plane of the cuprate superconductors. Within the linear spin wave approximation (LSW) of the Hamiltonian and the non-crossing approximation (NCA) for the hole self energies, thermal broadening of the hole spectral peaks is investigated with and without the contribution of optical phonons. We find the string excitations [15] to survive even for relatively strong electron-phonon coupling. Experimental angle-resolved photoemission spectroscopy (ARPES) results compare well with our calculations at finite temperature when we use strong electron-phonon coupling (γ). With vertex correction the agreement with experiment becomes possible even at a moderate value of γ.

Finite hole doping in a 2D t−J model and its extensions at T = 0 is studied using NCA. Dressed hole and magnon Green’s functions are obtained to analyze the hole energy bands/ Fermi surface topology and the magnon broadening and softening for the doped system. The doping-dependent staggered magnetization of the system is computed and the doping fraction up to which the staggered magnetization is non-zero is indicated. Contribution of the anomalous magnons has been reported as well.

We also study the dynamics of a hole in a 2D lattice in a stripe-ordered background. Within the same LSW and NCA treatment to the t−J Hamiltonian we obtain the different spin wave modes and hole Green’s functions of the superlattice structure. The hole spectra indicates the preference for the hole to be in the anti-phase domain wall rather than being within the antiferromagnetic block.